Part II

—Galois Theory

—

Year

2017201620152014201320122011201020092008200720062005

48

Paper 2, Section II

16I Galois Theory

(a) Define what it means for a finite field extension L of a field K to be separable. Showthat L is of the form K(α) for some α ∈ L.

(b) Let p and q be distinct prime numbers. Let L = Q(√p,√−q). Express L in the

form Q(α) and find the minimal polynomial of α over Q.

(c) Give an example of a field extension K 6 L of finite degree, where L is not of theform K(α). Justify your answer.

Paper 3, Section II

16I Galois Theory

(a) Let F be a finite field of characteristic p. Show that F is a finite Galois extensionof the field Fp of p elements, and that the Galois group of F over Fp is cyclic.

(b) Find the Galois groups of the following polynomials:

(i) t4 + 1 over F3.

(ii) t3 − t− 2 over F5.

(iii) t4 − 1 over F7.

Paper 1, Section II

17I Galois Theory

(a) Let K be a field and let f(t) ∈ K[t]. What does it mean for a field extension L ofK to be a splitting field for f(t) over K?

Show that the splitting field for f(t) over K is unique up to isomorphism.

(b) Find the Galois groups over the rationals Q for the following polynomials:

(i) t4 + 2t+ 2.

(ii) t5 − t− 1.

Part II, 2017 List of Questions

2017

49

Paper 4, Section II

17I Galois Theory

(a) State the Fundamental Theorem of Galois Theory.

(b) What does it mean for an extension L of Q to be cyclotomic? Show that a cyclotomicextension L of Q is a Galois extension and prove that its Galois group is Abelian.

(c) What is the Galois group G of Q(η) over Q, where η is a primitive 7th root ofunity? Identify the intermediate subfields M , with Q 6 M 6 Q(η), in terms of η,and identify subgroups of G to which they correspond. Justify your answers.

Part II, 2017 List of Questions [TURN OVER

2017

44

Paper 2, Section II

16H Galois Theory(a) Let K ⊆ L be a finite separable field extension. Show that there exist only

finitely many intermediate fields K ⊆ F ⊆ L.

(b) Define what is meant by a normal extension. Is Q ⊆ Q(√

1 +√7) a normal

extension? Justify your answer.

(c) Prove Artin’s lemma, which states: if K ⊆ L is a field extension, H is a finitesubgroup of AutK(L), and F := LH is the fixed field of H, then F ⊆ L is a Galoisextension with Gal(L/F ) = H.

Paper 3, Section II

16H Galois Theory(a) Let L be the 13th cyclotomic extension of Q, and let µ be a 13th primitive root of

unity. What is the minimal polynomial of µ over Q? What is the Galois group Gal(L/Q)?Put λ = µ+ 1

µ . Show that Q ⊆ Q(λ) is a Galois extension and find Gal(Q(λ)/Q).

(b) Define what is meant by a Kummer extension. Let K be a field of characteristiczero and let L be the nth cyclotomic extension of K. Show that there is a sequence ofKummer extensions K = F1 ⊆ F2 ⊆ · · · ⊆ Fr such that L is contained in Fr.

Paper 1, Section II

17H Galois Theory(a) Prove that if K is a field and f ∈ K[t], then there exists a splitting field L of f

over K. [You do not need to show uniqueness of L.]

(b) Let K1 and K2 be algebraically closed fields of the same characteristic. Showthat either K1 is isomorphic to a subfield of K2 or K2 is isomorphic to a subfield of K1.[For subfields Fi of K1 and field homomorphisms ψi : Fi → K2 with i = 1, 2, we say(F1, ψ1) 6 (F2, ψ2) if F1 is a subfield of F2 and ψ2|F1 = ψ1. You may assume the existenceof a maximal pair (F,ψ) with respect to the partial order just defined.]

(c) Give an example of a finite field extension K ⊆ L such that there existα, β ∈ L \K where α is separable over K but β is not separable over K.

Part II, 2016 List of Questions

2016

45

Paper 4, Section II

17H Galois Theory(a) Let f = t5 − 9t + 3 ∈ Q[t] and let L be the splitting field of f over Q. Show

that Gal(L/Q) is isomorphic to S5. Let α be a root of f . Show that Q ⊆ Q(α) is neithera radical extension nor a solvable extension.

(b) Let f = t26 + 2 and let L be the splitting field of f over Q. Is it true thatGal(L/Q) has an element of order 29? Justify your answer. Using reduction mod ptechniques, or otherwise, show that Gal(L/Q) has an element of order 3.

[Standard results from the course may be used provided they are clearly stated.]

Part II, 2016 List of Questions [TURN OVER

2016

44

Paper 3, Section II

16F Galois Theory

Let f ∈ Q[t] be of degree n > 0, with no repeated roots, and let L be a splittingfield for f .

(i) Show that f is irreducible if and only if for any α, β ∈ Rootf (L) there isφ ∈ Gal(L/Q) such that φ(α) = β.

(ii) Explain how to define an injective homomorphism τ : Gal(L/Q) → Sn. Find anexample in which the image of τ is the subgroup of S3 generated by (2 3). Find anotherexample in which τ is an isomorphism onto S3.

(iii) Let f(t) = t5 − 3 and assume f is irreducible. Find a chain of subgroups ofGal(L/Q) that shows it is a solvable group. [You may quote without proof any theoremsfrom the course, provided you state them clearly.]

Paper 4, Section II

17F Galois Theory

(i) Prove that a finite solvable extension K ⊆ L of fields of characteristic zero is aradical extension.

(ii) Let x1, . . . , x7 be variables, L = Q(x1, . . . , x7), and K = Q(e1, . . . , e7) where eiare the elementary symmetric polynomials in the variables xi. Is there an element α ∈ Lsuch that α2 ∈ K but α /∈ K? Justify your answer.

(iii) Find an example of a field extension K ⊆ L of degree two such that L 6= K(√α)

for any α ∈ K. Give an example of a field which has no extension containing an 11thprimitive root of unity.

Paper 2, Section II

17F Galois Theory

(i) State the fundamental theorem of Galois theory, without proof. Let L be asplitting field of t3 − 2 ∈ Q[t]. Show that Q ⊆ L is Galois and that Gal(L/Q) has asubgroup which is not normal.

(ii) Let Φ8 be the 8th cyclotomic polynomial and denote its image in F7[t] again byΦ8. Show that Φ8 is not irreducible in F7[t].

(iii) Let m and n be coprime natural numbers, and let µm = exp(2πi/m) andµn = exp(2πi/n) where i =

√−1. Show that Q(µm) ∩Q(µn) = Q.

Part II, 2015 List of Questions

2015

45

Paper 1, Section II

17F Galois Theory

(i) Let K ⊆ L be a field extension and f ∈ K[t] be irreducible of positive degree.Prove the theorem which states that there is a 1-1 correspondence

Rootf (L)←→ HomK

(K[t]

〈f〉 , L).

(ii) Let K be a field and f ∈ K[t]. What is a splitting field for f? What does itmean to say f is separable? Show that every f ∈ K[t] is separable if K is a finite field.

(iii) The primitive element theorem states that if K ⊆ L is a finite separable fieldextension, then L = K(α) for some α ∈ L. Give the proof of this theorem assuming K isinfinite.

Part II, 2015 List of Questions [TURN OVER

2015

44

Paper 4, Section II

18H Galois Theory(i) Let G be a finite subgroup of the multiplicative group of a field. Show that G is

cyclic.

(ii) Let Φn(X) be the nth cyclotomic polynomial. Let p be a prime not dividing n,and let L be a splitting field for Φn over Fp. Show that L has pm elements, where m isthe least positive integer such that pm ≡ 1 (mod n).

(iii) Find the degrees of the irreducible factors of X35 − 1 over F2, and the numberof factors of each degree.

Paper 3, Section II

18H Galois TheoryLet L/K be an algebraic extension of fields, and x ∈ L. What does it mean to say

that x is separable over K? What does it mean to say that L/K is separable?

Let K = Fp(t) be the field of rational functions over Fp.

(i) Show that if x is inseparable over K then K(x) contains a pth root of t.

(ii) Show that if L/K is finite there exists n > 0 and y ∈ L such that ypn= t and

L/K(y) is separable.

Show that Y 2+tY +t is an irreducible separable polynomial over the field of rationalfunctions K = F2(t). Find the degree of the splitting field of X4 + tX2 + t over K.

Paper 2, Section II

18H Galois TheoryDescribe the Galois correspondence for a finite Galois extension L/K.

Let L be the splitting field of X4 − 2 over Q. Compute the Galois group G of L/Q.For each subgroup of G, determine the corresponding subfield of L.

Let L/K be a finite Galois extension whose Galois group is isomorphic to Sn. Showthat L is the splitting field of a separable polynomial of degree n.

Part II, 2014 List of Questions

2014

45

Paper 1, Section II

18H Galois TheoryWhat is meant by the statement that L is a splitting field for f ∈ K[X]?

Show that if f ∈ K[X], then there exists a splitting field for f over K. Explain thesense in which a splitting field for f over K is unique.

Determine the degree [L : K] of a splitting field L of the polynomial f = X4−4X2+2over K in the cases (i) K = Q, (ii) K = F5, and (iii) K = F7.

Part II, 2014 List of Questions [TURN OVER

2014

42

Paper 4, Section II

18I Galois Theory(i) Let ζN = e2πi/N ∈ C for N > 1. For the cases N = 11, 13, is it possible to express

ζN , starting with integers and using rational functions and (possibly nested) radicals? Ifit is possible, briefly explain how this is done, assuming standard facts in Galois Theory.

(ii) Let F = C(X,Y,Z) be the rational function field in three variables over C, andfor integers a, b, c > 1 let K = C(Xa, Y b, Zc) be the subfield of F consisting of all rationalfunctions in Xa, Y b, Zc with coefficients in C. Show that F/K is Galois, and determineits Galois group. [Hint: For α, β, γ ∈ C×, the map (X,Y,Z) 7−→ (αX, βY, γZ) is anautomorphism of F .]

Paper 3, Section II

18I Galois TheoryLet p be a prime number and F a field of characteristic p. Let Frp : F → F be the

Frobenius map defined by Frp(x) = xp for all x ∈ F .

(i) Prove that Frp is a field automorphism when F is a finite field.

(ii) Is the same true for an arbitrary algebraic extension F of Fp? Justify youranswer.

(iii) Let F = Fp(X1, . . . ,Xn) be the rational function field in n variables wheren > 1 over Fp. Determine the image of Frp : F → F , and show that Frp makes F into anextension of degree pn over a subfield isomorphic to F . Is it a separable extension?

Paper 2, Section II

18I Galois TheoryFor a positive integer N , let Q(µN ) be the cyclotomic field obtained by adjoining

all N -th roots of unity to Q. Let F = Q(µ24).

(i) Determine the Galois group of F over Q.

(ii) Find all N > 1 such that Q(µN ) is contained in F .

(iii) List all quadratic and quartic extensions of Q which are contained in F , in theform Q(α) or Q(α, β). Indicate which of these fields occurred in (ii).

[Standard facts on the Galois groups of cyclotomic fields and the fundamentaltheorem of Galois theory may be used freely without proof.]

Part II, 2013 List of Questions

2013

43

Paper 1, Section II

18I Galois Theory(i) Give an example of a field F , contained in C, such that X4 + 1 is a product of

two irreducible quadratic polynomials in F [X]. Justify your answer.

(ii) Let F be any extension of degree 3 over Q. Prove that the polynomial X4 + 1is irreducible over F .

(iii) Give an example of a prime number p such that X4 + 1 is a product of twoirreducible quadratic polynomials in Fp[X]. Justify your answer.

[Standard facts on fields, extensions, and finite fields may be quoted without proof,as long as they are stated clearly.]

Part II, 2013 List of Questions [TURN OVER

2013

41

Paper 4, Section II

18H Galois TheoryLet F = C(X1, . . . ,Xn) be a field of rational functions in n variables over C, and

let s1, . . . , sn be the elementary symmetric polynomials:

sj :=∑

{i1,...,ij}⊂{1,...,n}Xi1 · · ·Xij ∈ F (1 6 j 6 n) ,

and let K = C(s1, . . . , sn) be the subfield of F generated by s1, . . . , sn. Let 1 6 m 6 n,and Y := X1 + · · · + Xm ∈ F . Let K(Y ) be the subfield of F generated by Y over K.Find the degree [K(Y ) : K].

[Standard facts about the fields F,K and Galois extensions can be quoted without proof,as long as they are clearly stated.]

Paper 3, Section II

18H Galois TheoryLet q = pf (f > 1) be a power of the prime p, and Fq be a finite field consisting of

q elements.

Let N be a positive integer prime to p, and Fq(µN ) be the cyclotomic extensionobtained by adjoining all Nth roots of unity to Fq. Prove that Fq(µN ) is a finite fieldwith qn elements, where n is the order of the element q mod N in the multiplicative group(Z/NZ)× of the ring Z/NZ.

Explain why what is proven above specialises to the following fact: the finite fieldFp for an odd prime p contains a square root of −1 if and only if p ≡ 1 (mod 4).

[Standard facts on finite fields and their extensions can be quoted without proof, aslong as they are clearly stated.]

Paper 2, Section II

18H Galois TheoryLet K,L be subfields of C with K ⊂ L.

Suppose that K is contained in R and L/K is a finite Galois extension of odd degree.Prove that L is also contained in R.

Give one concrete example of K,L as above with K 6= L. Also give an examplein which K is contained in R and L/K has odd degree, but is not Galois and L is notcontained in R.

[Standard facts on fields and their extensions can be quoted without proof, as longas they are clearly stated.]

Part II, 2012 List of Questions [TURN OVER

2012

42

Paper 1, Section II

18H Galois TheoryList all subfields of the cyclotomic field Q(µ20) obtained by adjoining all 20th roots

of unity to Q, and draw the lattice diagram of inclusions among them. Write all thesubfields in the form Q(α) or Q(α, β). Briefly justify your answer.

[The description of the Galois group of cyclotomic fields and the fundamental theorem ofGalois theory can be used freely without proof.]

Part II, 2012 List of Questions

2012

40

Paper 1, Section II

18H Galois TheoryLet K be a field.

(i) Let F and F ′ be two finite extensions ofK. When the degrees of these two extensionsare equal, show that every K-homomorphism F → F ′ is an isomorphism. Give anexample, with justification, of two finite extensions F and F ′ of K, which have thesame degrees but are not isomorphic over K.

(ii) Let L be a finite extension of K. Let F and F ′ be two finite extensions of L. Showthat if F and F ′ are isomorphic as extensions of L then they are isomorphic asextensions of K. Prove or disprove the converse.

Paper 2, Section II

18H Galois TheoryLet F = C(x, y) be the function field in two variables x, y. Let n > 1, and

K = C(xn + yn, xy) be the subfield of F of all rational functions in xn + yn and xy.

(i) Let K ′ = K(xn), which is a subfield of F . Show that K ′/K is a quadratic extension.

(ii) Show that F/K ′ is cyclic of order n, and F/K is Galois. Determine the Galoisgroup Gal(F/K).

Paper 3, Section II

18H Galois TheoryLet n > 1 and K = Q(µn) be the cyclotomic field generated by the nth roots of

unity. Let a ∈ Q with a 6= 0, and consider F = K( n√a).

(i) State, without proof, the theorem which determines Gal(K/Q).

(ii) Show that F/Q is a Galois extension and that Gal(F/Q) is soluble. [When usingfacts about general Galois extensions and their generators, you should state themclearly.]

(iii) When n = p is prime, list all possible degrees [F : Q], with justification.

Part II, 2011 List of Questions

2011

41

Paper 4, Section II

18H Galois TheoryLet K be a field of characteristic 0, and let P (X) = X4 + bX2 + cX + d be an

irreducible quartic polynomial over K. Let α1, α2, α3, α4 be its roots in an algebraicclosure of K, and consider the Galois group Gal(P ) (the group Gal(F/K) for a splittingfield F of P over K) as a subgroup of S4 (the group of permutations of α1, α2, α3, α4).

Suppose that Gal(P ) contains V4 = {1, (12)(34), (13)(24), (14)(23)}.

(i) List all possible Gal(P ) up to isomorphism. [Hint: there are 4 cases, with orders 4,8, 12 and 24.]

(ii) Let Q(X) be the resolvent cubic of P , i.e. a cubic in K[X] whose roots are−(α1 + α2)(α3 + α4), −(α1 + α3)(α2 + α4) and −(α1 + α4)(α2 + α3). Constructa natural surjection Gal(P ) → Gal(Q), and find Gal(Q) in each of the four casesfound in (i).

(iii) Let ∆ ∈ K be the discriminant of Q. Give a criterion to determine Gal(P ) in termsof ∆ and the factorisation of Q in K[X].

(iv) Give a specific example of P where Gal(P ) is abelian.

Part II, 2011 List of Questions [TURN OVER

2011

39

Paper 1, Section II

18H Galois TheoryLet Fq be a finite field with q elements and Fq its algebraic closure.

(i) Give a non-zero polynomial P (X) in Fq [X1, . . . ,Xn] such that

P (α1, . . . , αn) = 0 for all α1, . . . , αn ∈ Fq .

(ii) Show that every irreducible polynomial P (X) of degree n > 0 in Fq[X] can be

factored in Fq[X] as(X−α

)(X−α q

)(X−α q 2) · · ·

(X−α q n−1)

for some α ∈ Fq . Whatis the splitting field and the Galois group of P over Fq?

(iii) Let n be a positive integer and Φn(X) be the n-th cyclotomic polynomial.Recall that if K is a field of characteristic prime to n , then the set of all roots of Φn in Kis precisely the set of all primitive n-th roots of unity in K . Using this fact, prove that ifp is a prime number not dividing n , then p divides Φn(x) in Z for some x ∈ Z if and onlyif p = an+ 1 for some integer a . Write down Φn explicitly for three different values of nlarger than 2, and give an example of x and p as above for each n .

Paper 2, Section II

18H Galois Theory(1) Let F = Q( 3

√5 ,

√5 , i). What is the degree of F/Q? Justify your answer.

(2) Let F be a splitting field of X4 − 5 over Q . Determine the Galois groupGal(F/Q). Determine all the subextensions of F/Q , expressing each in the form Q(x) orQ(x, y) for some x, y ∈ F .

[Hint: If an automorphism ρ of a field X has order 2 , then for every x ∈ X the elementx+ ρ(x) is fixed by ρ .]

Part II, 2010 List of Questions [TURN OVER

2010

40

Paper 3, Section II

18H Galois TheoryLet K be a field of characteristic 0 . It is known that soluble extensions of K

are contained in a succession of cyclotomic and Kummer extensions. We will refine thisstatement.

Let n be a positive integer. The n-th cyclotomic field over a field K is denoted by K(µn).Let ζn be a primitive n-th root of unity in K(µn).

(i) Write ζ 3 ∈ Q(µ3), ζ 5 ∈ Q(µ5) in terms of radicals. Write Q(µ3)/Q and Q(µ5)/Qas a succession of Kummer extensions.

(ii) Let n > 1, and F := K(ζ 1 , ζ 2 , . . . , ζn−1). Show that F (µn)/F can be writtenas a succession of Kummer extensions, using the structure theorem of finite abelian groups(in other words, roots of unity can be written in terms of radicals). Show that every solubleextension of K is contained in a succession of Kummer extensions.

Paper 4, Section II

18H Galois TheoryLet K be a field of characteristic 6= 2, 3 , and assume that K contains a primitive

cubic root of unity ζ. Let P ∈ K[X] be an irreducible cubic polynomial, and let α, β, γbe its roots in the splitting field F of P over K . Recall that the Lagrange resolvent x ofP is defined as x = α+ ζβ + ζ 2 γ .

(i) List the possibilities for the group Gal(F/K), and write out the set{σ(x) | σ ∈ Gal(F/K)} in each case.

(ii) Let y = α + ζγ + ζ 2β . Explain why x 3, y 3 must be roots of a quadraticpolynomial in K[X] . Compute this polynomial for P = X 3 + bX + c , and deduce thecriterion to identify Gal(F/K) through the element −4b 3 − 27c 2 of K .

Part II, 2010 List of Questions

2010

43

Paper 1, Section II

18H Galois Theory

Define a K-isomorphism, ϕ : L → L′, where L, L′ are fields containing a field K,

and define AutK(L).

Suppose α and β are algebraic over K. Show that K(α) and K(β) are K-isomorphic

via an isomorphism mapping α to β if and only if α and β have the same minimal

polynomial.

Show that AutKK(α) is finite, and a subgroup of the symmetric group Sd, where d

is the degree of α.

Give an example of a field K of characteristic p > 0 and α and β of the same

degree, such that K(α) is not isomorphic to K(β). Does such an example exist if K is

finite? Justify your answer.

Paper 2, Section II

18H Galois Theory

For each of the following polynomials over Q, determine the splitting field K and

the Galois group G.

(1) x4 − 2x2 − 25.

(2) x4 − 2x2 + 25.

Paper 3, Section II

18H Galois Theory

Let K = Fp(x), the function field in one variable, and let G = Fp. The group G acts

as automorphisms of K by σa(x) = x+ a. Show that KG = Fp(y), where y = xp − x.

[State clearly any theorems you use.]

Is K/KG a separable extension?

Now let

H =

{(d a

0 1

): a ∈ Fp, d ∈ F∗p

}

and let H act on K by

(d a

0 1

)x = dx+a. (The group structure on H is given by matrix

multiplication.) Compute KH . Describe your answer in the form Fp(z) for an explicit

z ∈ K.

Is KG/KH a Galois extension? Find the minimum polynomial for y over the

field KH .

Part II, 2009 List of Questions [TURN OVER

2009

44

Paper 4, Section II

18H Galois Theory

(a) Let K be a field. State what it means for ξn ∈ K to be a primitive nth root of

unity.

Show that if ξn is a primitive nth root of unity, then the characteristic of K does

not divide n. Prove any theorems you use.

(b) Determine the minimum polynomial of a primitive 10th root of unity ξ10 over Q.

Show that√5 ∈ Q(ξ10).

(c) Determine F3(ξ10), F11(ξ10), F19(ξ10).

[Hint: Write a necessary and sufficient condition on q for a finite field Fq to contain a

primitive 10th root of unity.]

Part II, 2009 List of Questions

2009

41

1/II/18H Galois Theory

Find the Galois group of the polynomial f(x) = x4 + x3 + 1 over

(i) the finite field F2, (ii) the finite field F3,

(iii) the finite field F4, (iv) the field Q of rational numbers.

[Results from the course which you use should be stated precisely.]

2/II/18H Galois Theory

(i) Let K be a field, θ ∈ K, and n > 0 not divisible by the characteristic. Supposethat K contains a primitive nth root of unity. Show that the splitting field of xn − θ hascyclic Galois group.

(ii) Let L/K be a Galois extension of fields and ζn denote a primitive nth root ofunity in some extension of L, where n is not divisible by the characteristic. Show thatAut(L(ζn)/K(ζn)) is a subgroup of Aut(L/K).

(iii) Determine the minimal polynomial of a primitive 6th root of unity ζ6 over Q.

Compute the Galois group of x6 + 3 ∈ Q[x].

3/II/18H Galois Theory

Let L/K be a field extension.

(a) State what it means for α ∈ L to be algebraic over K, and define its degreedegK(α). Show that if degK(α) is odd, then K(α) = K(α2).

[You may assume any standard results.]

Show directly from the definitions that if α, β ∈ L are algebraic over K, then sotoo is α+ β.

(b) State what it means for α ∈ L to be separable over K, and for the extensionL/K to be separable.

Give an example of an inseparable extension L/K.

Show that an extension L/K is separable if L is a finite field.

Part II 2008

2008

42

4/II/18H Galois Theory

Let L = C(z) be the function field in one variable, n > 0 an integer, and ζn = e2πi/n.

Define σ, τ : L→ L by the formulae

(σf)(z) = f(ζnz), (τf)(z) = f(1/z),

and let G = 〈σ, τ〉 be the group generated by σ and τ .

(i) Find w ∈ C(z) such that LG = C(w).

[You must justify your answer, stating clearly any theorems you use.]

(ii) Suppose n is an odd prime. Determine the subgroups of G and the correspond-ing intermediate subfields M , with C(w) ⊆M ⊆ L.

State which intermediate subfields M are Galois extensions of C(w), and for theseextensions determine the Galois group.

Part II 2008

2008

40

1/II/18F Galois Theory

Let L/K/M be field extensions. Define the degree [K : M ] of the field extensionK/M , and state and prove the tower law.

Now let K be a finite field. Show #K = pn, for some prime p and positive integern. Show also that K contains a subfield of order pm if and only if m|n.

If f ∈ K[x] is an irreducible polynomial of degree d over the finite fieldK, determineits Galois group.

2/II/18F Galois Theory

Let L = K(ξn), where ξn is a primitive nth root of unity and G = Aut(L/K).Prove that there is an injective group homomorphism χ : G→ (Z/nZ)∗.

Show that, ifM is an intermediate subfield of K(ξn)/K, thenM/K is Galois. Statecarefully any results that you use.

Give an example where G is non-trivial but χ is not surjective. Show that χ issurjective when K = Q and n is a prime.

Determine all the intermediate subfields M of Q(ξ7) and the automorphism groupsAut(Q (ξ7)/M). Write the quadratic subfield in the form Q(

√d) for some d ∈ Q.

3/II/18F Galois Theory

(i) Let K be the splitting field of the polynomial x4 − 3 over Q. Describe the field K,the Galois group G = Aut(K/Q), and the action of G on K.

(ii) Let K be the splitting field of the polynomial x4 + 4x2 + 2 over Q. Describe thefield K and determine Aut(K/Q).

4/II/18F Galois Theory

Let f(x) ∈ K[x] be a monic polynomial, L a splitting field for f , α1, . . . , αn theroots of f in L. Let 4(f) =

∏i<j(αi−αj)

2 be the discriminant of f . Explain why 4(f) is

a polynomial function in the coefficients of f , and determine 4(f) when f(x) = x3+px+q.

Compute the Galois group of the polynomial x3 − 3x+ 1 ∈ Q[x].

Part II 2007

2007

38

1/II/18H Galois Theory

Let K be a field and f a separable polynomial over K of degree n. Explain whatis meant by the Galois group G of f over K. Show that G is a transitive subgroup of Sn

if and only if f is irreducible. Deduce that if n is prime, then f is irreducible if and onlyif G contains an n-cycle.

Let f be a polynomial with integer coefficients, and p a prime such that f , thereduction of f modulo p, is separable. State a theorem relating the Galois group of f overQ to that of f over Fp.

Determine the Galois group of the polynomial x5 − 15x− 3 over Q.

2/II/18H Galois Theory

Write an essay on ruler and compass construction.

3/II/18H Galois Theory

Let K be a field and m a positive integer, not divisible by the characteristic of K.Let L be the splitting field of the polynomial Xm − 1 over K. Show that Gal(L/K) isisomorphic to a subgroup of (Z/mZ)∗.

Now assume that K is a finite field with q elements. Show that [L : K] is equal tothe order of the residue class of q in the group (Z/mZ)∗. Hence or otherwise show thatthe splitting field of X11 − 1 over F4 has degree 5.

4/II/18H Galois Theory

Let K be a field of characteristic different from 2.

Show that if L/K is an extension of degree 2, then L = K(x) for some x ∈ L suchthat x2 = a ∈ K. Show also that if L′ = K(y) with 0 6= y2 = b ∈ K then L and L′ areisomorphic (as extensions of K) if and only b/a is a square in K.

Now suppose that F = K(x1, . . . , xn) where 0 6= x2i = ai ∈ K. Show that F/Kis a Galois extension, with Galois group isomorphic to (Z/2Z)m for some m 6 n. Byconsidering the subgroups of Gal(F/K), show that if K ⊂ L ⊂ F and [L : K] = 2, then

L = K(y) where y =∏

i∈I

xi for some subset I ⊂ {1, . . . , n}.

Part II 2006

2006

36

1/II/18G Galois Theory

Let L/K be a field extension. State what it means for an element x ∈ L to bealgebraic over K. Show that x is algebraic over K if and only if the field K(x) is finitedimensional as a vector space over K.

State what it means for a field extension L/K to be algebraic. Show that, if M/Lis algebraic and L/K is algebraic, then M/K is algebraic.

2/II/18G Galois Theory

Let K be a field of characteristic 0 containing all roots of unity.

(i) Let L be the splitting field of the polynomial Xn − a where a ∈ K. Show thatthe Galois group of L/K is cyclic.

(ii) Suppose that M/K is a cyclic extension of degree m over K. Let g be agenerator of the Galois group and ζ ∈ K a primitive m-th root of 1. By considering theresolvent

R(w) =m−1∑

i=0

gi(w)

ζi

of elements w ∈ M , show that M is the splitting field of a polynomial Xm − a for somea ∈ K.

3/II/18G Galois Theory

Find the Galois group of the polynomial

x4 + x+ 1

over F2 and F3. Hence or otherwise determine the Galois group over Q.

[Standard general results from Galois theory may be assumed.]

4/II/18G Galois Theory

(i) Let K be the splitting field of the polynomial

x4 − 4x2 − 1

over Q. Show that [K : Q] = 8, and hence show that the Galois group of K/Q is thedihedral group of order 8.

(ii) Let L be the splitting field of the polynomial

x4 − 4x2 + 1

over Q. Show that [L : Q] = 4. Show that the Galois group of L/Q is C2 × C2.

Part II 2005

2005